A322577 a(n) = Sum_{d|n} psi(n/d) * phi(d).

1, 4, 6, 11, 10, 24, 14, 28, 26, 40, 22, 66, 26, 56, 60, 68, 34, 104, 38, 110, 84, 88, 46, 168, 74, 104, 102, 154, 58, 240, 62, 160, 132, 136, 140, 286, 74, 152, 156, 280, 82, 336, 86, 242, 260, 184, 94, 408, 146, 296, 204, 286, 106, 408, 220, 392, 228, 232, 118, 660

Offset: 1

Author

Ilya Gutkovskiy, Aug 29 2019

Keywords

• nonn
• mult
• easy

Comments

Dirichlet convolution of Dedekind psi function (A001615) with Euler totient function (A000010).

Dirichlet convolution of A008966 with A018804.

Dirichlet convolution of A038040 with A271102.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Dedekind Function.
• Eric Weisstein's World of Mathematics, Totient Function.

Crossrefs

Cf. A000010, A001615, A008966, A018804, A038040, A271102.

Cf. A327251 (inverse Möbius transform), A347092 (Dirichlet inverse), A347093 (sum with it), A347135.

Programs

Maple

f:= proc(n) local t;
  mul((t[2]+1)*t[1]^t[2] - (t[2]-1)*t[1]^(t[2]-2), t = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2019

Mathematica

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e + 1)*p^e - (e - 1)*p^(e - 2); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

PARI

seq(n) = {dirmul(vector(n, n, eulerphi(n)), vector(n, n, n * sumdivmult(n, d, issquarefree(d)/d)))} \\ Andrew Howroyd, Aug 29 2019

PARI

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d)); \\ Antti Karttunen, Apr 03 2022

Formula

Dirichlet g.f.: zeta(s-1)^2 / zeta(2*s).

a(p) = 2*p, where p is prime.

Sum_{k=1..n} a(k) ~ 45*n^2*(2*Pi^4*log(n) - Pi^4 + 4*gamma*Pi^4 - 360*zeta'(4)) / (2*Pi^8), where gamma is the Euler-Mascheroni constant A001620 and for zeta'(4) see A261506. - Vaclav Kotesovec, Aug 31 2019

a(p^k) = (k+1)*p^k - (k-1)*p^(k-2) where p is prime. - Robert Israel, Sep 01 2019

a(n) = Sum_{k=1..n} psi(gcd(n,k)). - Ridouane Oudra, Nov 29 2019

a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021